Definition 2 (Supermodular Game [1]}) 
A game with a set of players \(\mathcal {Y}\) , strategy space \((X_v)_{v\in \mathcal {Y}}\)  and payoff function \((U_v)_{v\in \mathcal {Y}}\)  (denoted as \(\mathcal {G}=[\mathcal {Y},(X_v)_{v\in \mathcal {Y}},(U_v)_{v\in \mathcal {Y}}]\) ) is a supermodular game if for each player \(v\in \mathcal {Y}\) :
